Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $y \neq 0$. $t = \dfrac{y^2 - 2y - 35}{y + 7} \div \dfrac{y + 5}{y + 7} $
Dividing by an expression is the same as multiplying by its inverse. $t = \dfrac{y^2 - 2y - 35}{y + 7} \times \dfrac{y + 7}{y + 5} $ First factor the quadratic. $t = \dfrac{(y + 5)(y - 7)}{y + 7} \times \dfrac{y + 7}{y + 5} $ Then multiply the two numerators and multiply the two denominators. $t = \dfrac{ (y + 5)(y - 7) \times (y + 7) } { (y + 7) \times (y + 5) } $ $t = \dfrac{ (y + 5)(y - 7)(y + 7)}{ (y + 7)(y + 5)} $ Notice that $(y + 7)$ and $(y + 5)$ appear in both the numerator and denominator so we can cancel them. $t = \dfrac{ \cancel{(y + 5)}(y - 7)(y + 7)}{ (y + 7)\cancel{(y + 5)}} $ We are dividing by $y + 5$ , so $y + 5 \neq 0$ Therefore, $y \neq -5$ $t = \dfrac{ \cancel{(y + 5)}(y - 7)\cancel{(y + 7)}}{ \cancel{(y + 7)}\cancel{(y + 5)}} $ We are dividing by $y + 7$ , so $y + 7 \neq 0$ Therefore, $y \neq -7$ $t = y - 7 ; \space y \neq -5 ; \space y \neq -7 $